Baby's head not engaged

I am 40+2 and at my last apt baby's head was still not engaged and she was sitting high. I am only 1/2 cm dilated, 2nd week in a row, but cervix has moved forward and down, but still thick. If she doesn't come on her own by Sunday I will be induced. Has anyone had a baby that wouldn't engage and was still able to have a vaginal birth after induction? I'm worried it's going to lead to a c section.